Power-law Rate Research Articles

Simplifying population balance models to promote broader use in industry

The approach outlined in this paper takes advantage of the existence of stationary states and similarity solutions to reduce a one-dimensional population balance model with particle volume as the internal dimension from a partial integro-differential equation to a single ordinary differential equation for the evolution of the mean size. The mechanisms of accretional growth, collisional growth and breakage, either individually or in all possible combinations, are considered. Power-law rate kernels are employed and a self-similar fragment distribution is assumed for the breakage term. When expressed in terms of the evolution of the scaled number-mean particle volume, the model parameters consist of kernel orders and the characteristic times associated with each mechanism. Equations for the scaled moments of the stationary states and similarity solutions are exhibited for all possible mechanism combinations as are equations for the trajectories of scaled number-mean particle volume. Stability of the stationary states and similarity solutions is addressed in an appendix. It should be clear that when this level of simplification is possible, the subject of population balance modeling becomes considerably easier to teach while its use becomes much easier to promote in industry. The use of this simplified approach for extraction of model parameters from experiment is demonstrated in an example, followed by a brief discussion of using the results for scale up. To close out the paper, the same approach, but now in two internal dimensions (particle volume and surface area), is used to demonstrate the capability of the model to unify a set of operating conditions for carbon black manufacture across multiple grades and two different types of synthesis reactors. This is done by extracting the characteristic coalescence time at multiple temperatures and correlating the results in an Arrhenius plot. The model matches known values of primary particle size, and predicts a key indicator of the product’s fractal aggregate structure, namely the number of primary particles per aggregate. The model results are then pushed to yield insight into the nature of the possible collision and coalescence mechanisms.

Effects of heat/mass transfer limitations and process exothermicity on the kinetic parameters of the devolatilization and oxidation reactions of wood chars

Thermogravimetric curves of wood char oxidation consist of devolatilization and oxidation, described with a linear and a power-law rate reaction, respectively. The lack of measurement and control of the sample temperature in standard commercial thermogravimetric systems (STGA) can introduce serious deviations from the heating program, originated from heat and mass transfer limitations and reaction exothermicity. This study aims at quantifying the consequent effects on the estimated kinetic constants for beech, fir and spruce wood chars. A first set of measurements is based on a special purpose designed, homemade thermo-analyzer (HMTGA), permitting for an accurate sample temperature control and a rigorous kinetic regime. The second set makes use of a standard commercial thermobalance (STGA). It is found that for heating rates up to 10 K/min spikes in the rate curves disappear for 1 mg samples and the deviations on the activation energies become null or very small (maximum around 1–1.5%), compared with the HMTGA data. However, the order of the oxidation reaction is always lower (by about 15%). Improvements are not observed following further sample mass reduction.

A distributed domain model coupling open channel flow and groundwater flow to quantify the impact of lateral hydrologic exchange on hydrograph

• A distributed-domain coupling model is developed to evaluate the interaction between hydrologic exchange flows and river hydrodynamics. • The buffering effect of lateral HEFs can dominate flood wave patterns. • The falling limb (with a power-law rate) of hydrographs is linked to spatially heterogeneous hydraulic properties. Hydrologic exchange flows (HEFs) of river-aquifer systems are known to affect water flow, but the quantitative influences of lateral HEFs and the riparian zone’s hydraulic conductivity ( K ) distribution on stream fluxes remain obscure under varying hydrologic conditions. To fill this knowledge gap, this study proposed a physical based, distributed domain, coupled (open channel and groundwater) flow model (DDCM) to quantify the effect of lateral HEFs on hydrograph characteristics, including especially the peak discharge and tailing decay which are practically important. Numerical experiments showed that (1) the interaction between the lateral HEFs and river hydrodynamics reduced the peak discharge and flood flow rates, (2) a heterogeneous K field of the riparian zone generated multi-rate HEFs (which then changed flow response) represented by the hydrographs with various declining rates (varying from exponential to power-law), significantly expanding the flow process, and (3) the probability density function of K also affected the tailing and peak of the hydrograph. A preliminary test showed that the DDCM captured the overall pattern of hydrographs observed from a catchment in the Wadi Ahin West, Oman. This study, therefore, provided a model-based quantification of the mechanisms and factors of the lateral HEFs affecting the hydrograph pattern in flood events, and further applications are needed to test the applicability of the DDCM in capturing real-world hydrographs affected by the lateral HEFs.

Deep hydrodesulfurization of gas oils with high sulfur content: Experiment and kinetic modeling

The hydrodesulfurization of straight-run gas oils (SRGO-HS) with high content of sulfur (about 2 wt%) and its mixture with light coker gas oil (CGO) has been studied over CoMo/Al2O3 catalyst using a trickle bed down-flow reactor at the 335–365 °C, H2/HC − 300 Nm3/m3, LHSV − 0.8–2.5 h−1 and 3.5–5.1 MPa. The possibility of ultra-low sulfur diesel (<10 ppmS) production using feedstocks with sulfur content exceeding 2 wt% was demonstrated under conditions simulating industrial one. To predict the effect of process parameters on the amount of sulfur in products four kinetic models with different insight into the reactivity of individual S-containing compounds were considered. The first model is simplified kinetics based on power law rate equations, taking into account overall sulfur content, the second group includes LHHW type rate equations. The models 3 and 4 take into consideration the difference in the reactivity of sulfur-containing components. Model 3 classifies the total sulfur into two subgroups Se (easy reactive compounds) and Sr (refractory sulfur compounds - dibenzothiophene and its derivatives). In Model 4 the group Sr is divided into three more subgroups. Kinetic rate parameters i.e. frequency factors, apparent activation energies, and adsorption enthalpies, were estimated by fitting the experimental data obtained using the SRGO-HS/CGO feedstock to the models. All kinetic models predicted the experimentally determined conversions of sulfur with a relatively good accuracy (±20%). The kinetic models were verified by comparing the simulation results with the experimental data obtained using the other oil fractions (SRGO-LS) with lower content of sulfur (1.05 wt%). A better agreement was observed between the experiments and prediction by models 3 and 4.

On Accompanying Measures and Asymptotic Expansions in the B. V. Gnedenko Limit Theorem

We propose a sequence of accompanying laws in the B.,V. Gnedenko limit theorem for maxima of independent random variables with distributions lying in the Gumbel max domain of attraction. We show that this sequence provides a power-law convergence rate, whereas the Gumbel distribution provides only the logarithmic rate. As examples, we consider in detail the classes of Weibull and log-Weibull type distributions. For the entire Gumbel max domain of attraction, we propose a scale of classes of distributions that includes these two classes as a starting point.

A global kinetic model for the oxidative dehydrogenation of ethane over mixed metal oxide catalysts at supra-ambient pressures

Oxidative dehydrogenation of ethane (ODHE), unlike traditional steam cracking processes, can potentially be used to produce ethene at supra-ambient pressures, thereby reducing reactor footprint and alleviating (de)compression energy requirements. Global kinetic models that capture kinetic features of both desired and undesired reactions over an extended range of reactant and product pressures are lacking despite the clear reliance of comparative reactor design assessments on such models. We report herein a global kinetic model that accounts for the rates (between 603 and 703 K) of all 6 of the prevalent reaction network steps, extends up to 6 bar total pressure, and explains a broad set of differential and integral kinetic features measured over MoVTeNbOx catalysts. H2O2-mediated dissolution procedures enable low-temperature measurements which under the extended pressure ranges used in this study evidence significant coverages of reduced sites the contributions of which can be interpreted as being determined by ethane to oxygen molar ratios. Explaining measured kinetic features require invoking an oxygen pool present in quasi-equilibrium with gas phase oxygen that is distinct in identity from lattice oxygens, only the latter of which are wholly responsible for hydrogen abstraction steps in turnovers producing ethene, not COx. We demonstrate how a simplified global kinetic model that employs power law rate expressions for undesired reactions and excludes product inhibitory effects for the entirety of the reaction network is sufficient to explain both co-feed data as well as differential and integral features evaluated in the absence of product co-feeds. The proposed kinetic model can be employed in comparative assessments of high-pressure ODHE reactor configurations operating non-isothermally, especially those carrying a high sensitivity to contributions from highly exothermic total oxidation reactions.

Oxidation and electrochemical characterization of oxide formed on Ti-Al-Zr in a neutral steam environment

The study focuses on the oxidation and the electrochemical behavior of the oxide formed on an alpha titanium alloy, Ti-Al-Zr, in a neutral steam environment. The alloy was exposed in a static autoclave at 400 °C, 10 MPa for durations up to 90 days. The oxidation kinetics and the rate laws were established by the discontinuous gravimetric method. The characterization of the oxide was done by scanning electron microscopy-energy dispersive spectroscopy (SEM-EDS), glow discharge optical emission spectroscopy (GDOES) and Raman spectroscopy. The electrochemical characterization of the high-temperature oxide film was done by a potentiodynamic polarization experiment and ex-situ electrochemical impedance spectroscopy (EIS) measurement. The defect density of the oxide was calculated using the Mott-Schottky (M-S) analysis. A physical model was developed based on the impedance response, within the framework of point defect model (PDM) theory. The oxidation kinetics followed a nearly fourth power rate law followed by a linear rate law. The oxide consisted of an inner barrier layer and an outer layer containing crystallized grains. The transition in the oxidation kinetics was attributed to the recrystallization of the outer layer. The defect density for all the oxidized specimens was of the order of 1021 cm−3. The EIS study, in conjunction with the PDM, established that the oxidation kinetic was controlled by both titanium interstitials and oxygen vacancies. A correlation between the oxidation kinetics and the kinetics of the interfacial reactions was established and the oxide growth mechanism in the steam environment is discussed.

Insights into the intrinsic kinetics for efficient hydrodesulfurization of 4,6-dimethyldibenzothiophene over mesoporous CoMoS2/ZSM-5

The intrinsic kinetics of hydrodesulfurization (HDS) of 4,6-dimethyldibenzothiophene (4,6-DMDBT) over mesoporous CoMoS2/ZSM-5 (CoMoS2/MZSM-5) was investigated by comparing the commercial CoMoS2/γ-Al2O3. A pseudo-first-order kinetic model considering the direct desulfurization (DDS) and two detailed partial and full hydrogenation pathways was formulated with simple power law rate equations. The apparent rate constants of five lumped reactions on CoMoS2/MZSM-5 are 2–4 times larger than those of CoMoS2/γ-Al2O3. With both catalyst system, the pre-exponential factor plays a dominant role in the reaction rate of DDS pathway showing equally high activation energies. The apparent activation energies show a critical effect on the reaction rate of two hydrogenation and subsequent desulfurization reactions. Furthermore, the Delplot analysis was applied to identify the reaction sequence of product formation and verify the reaction network. The HDS performance and reaction rates of individual reaction steps can be rationally linked to the morphology and available content of CoMoS phase as identified by HRTEM and XPS characterization. Thus, multi-stacked MoS2 nanocrystallites on CoMoS2/MZSM-5 provide more available sulfur vacancies to facilitate the DDS pathway, and catalyze efficiently the hydrogenation and subsequent desulfurization with the synergy of brim sites and sulfur vacancies.

Novel Cu-doped zeolitic imidazolate framework-8 membranes supported on copper foam for highly efficient catalytic wet peroxide oxidation of phenol

Cu-doped zeolitic imidazolate framework-8 (ZIF-8) membranes supported on three-dimensional copper foam were prepared for the first time and used for the catalytic wet peroxide oxidation of phenol in a fixed bed reactor. The properties of the new porous composite materials were studied by scanning electron microscope, energy-dispersive spectroscopy, Brunauer-Emmett-Teller surface area analysis, X-ray diffraction, thermal gravimetric analysis, Fourier-transform infrared, and X-ray photoelectron spectroscopy. The ZIF-8 membrane doped with 5% copper exhibited the best catalytic performance (the removal efficiency of phenol and H2O2 reached 100% and 95%, respectively). Reusability and stability experiments showed that a consistent complete phenol conversion and high H2O2 conversions of more than 94% were achieved using the 5% Cu-doped ZIF-8 membrane. The effects of reaction temperature, feed flow rate, and catalyst bed height on the catalytic performance were investigated to obtain the optimal reaction conditions. The material has the more economical characteristics of short reaction time, low metal loading, continuous operation, easy recycling, and good stability. The kinetics of the catalytic reaction of the Cu-doped ZIF-8 membrane were analyzed using the Power-rate Law Kinetic model. The experimental results show that the reaction is a first-order reaction, and the activation energy (Ea) of the catalyzed oxidation reaction is 66.4 kJ/mol. The initial oxidation rate equation (−rA0 = 3.24 × 108e−66400/RT CA) was obtained for Cu-doped ZIF-8 membrane–catalyzed phenol degradation.

Non-Newtonian and viscoplastic models of a vertically aligned thick liquid film draining due to gravity

We consider theoretically the two-dimensional flow in a vertically aligned thick liquid film supported at the top and bottom by wire frames. The film gradually thins as the liquid drains due to gravity. We focus on investigating the influence of non-Newtonian and viscoplastic effects, such as shear thinning and yield stress, on the draining and thinning of the liquid film, important in metallic and polymeric melt films. Lubrication theory is employed to derive coupled equations for a generalized Newtonian liquid describing the evolution of the film's thickness and the extensional flow speed. We use the non-Newtonian (power-law and Carreau) and viscoplastic (Bingham and Herschel–Bulkley) constitutive laws to describe the flow rheology. Numerical solutions combined with asymptotic solutions predict the late-time power-law thinning rate of the middle section of the film. For a Newtonian liquid, a new power law thinning rate of t−2.25 is identified. This is in comparison with a thinning rate of t−2 predicted for a thin Newtonian liquid film neglecting gravity, suggesting a weak dependence on gravity for the drainage of thicker films. For a non-Newtonian and viscoplastic liquid, varying the power law index and the yield stress influences the timescale of the thinning, but has weak dependence on the late-time thinning rate relative to the Newtonian thinning rate. The shortcomings of the power-law model are exposed when the shear rate is low and these are resolved using the Carreau model.

Influence of grain size on the high-temperature creep behaviour of M5Framatome1 zirconium alloy under vacuum

The effect of grain size on the viscoplastic behaviour of M5Framatome zirconium alloy thin sheets was investigated at high temperature under uniaxial tension, using a variety of equiaxed microstructures with controlled grain sizes. In the α phase domain, a Coble diffusional creep regime and a dislocation creep regime were observed, in agreement with the literature. A negative sensitivity of the strain rate to temperature was highlighted in the upper part of the α+β two-phase temperature range, consistently with the literature. For the first time, a linear creep regime was evidenced in the β phase domain. In this regime, a sensitivity of the strain rate on the third power of the grain size is observed, suggesting a Coble regime with diffusion along grain boundaries. Modelling with multi-mechanism Norton power-law rate equations, including dependence on grain size and a homogenous strain-rate assumption (Taylor model), enabled to satisfactorily reproduce the experimental results over the 700–1100 °C temperature range, especially the negative sensitivity of the strain rate to temperature between 880 and 930 °C. Very good agreement was obtained with a second order self-consistent and full field homogenization schemes.

Three different models to evaluate water discharge: An application to a river section at Vinh Tuy location in the Lo river basin, Vietnam

This study presents three different models, namely power-law rating curve, one-dimensional lateral distribution method (1D–LDM), and gated recurrent network (GRU) model that can be applied to evaluate water discharge from water surface elevation time-series in a river cross-section for a long time period. A river section at Vinh Tuy location on the Lo river basin (Vietnam) is used to demonstrate the models. Appropriate values of modelling parameters are carefully determined using (i) the daily observed discharge values collected in the period from 2012 to 2018 and (ii) five error estimates for quantitatively assessing the agreement between estimated and observed water discharges. The results showed that all three models reproduced very well the observed discharge values, with root mean square error and mean absolute error, as well as mean error of discharge, are only about 5.5% of the maximum value of discharge monitoring in the studied cross-section, while Nash–Sutcliffe efficiency and Pearson’s correlation coefficient are greater than 0.89. The models are then applied to evaluate discharge values in the studied cross-section for the period from 1972 to 2011, revealing that statistical indicators, i.e. mean value, standard derivation, and covariance of estimated water discharge, are consistent with those obtained from the observations. Among three investigated models, the GRU model was finally proved to be the best one, providing even better results than the 1D-LDM and power-law rating curve.

Deactivation kinetics of γ-Al2O3 catalyst in methanol dehydration to dimethyl ether

The kinetics of deactivation of γ-alumina catalyst in dehydration of methanol to dimethyl ether (DME) was studied using integral method of analysis. A power-law rate expression was used for dehydration reaction and an independent kinetics for catalyst deactivation. Catalytic test runs were performed in a fixed-bed integral reactor in the temperature range of 250–300 °C under kinetically controlled reaction conditions. A second-order independent deactivation kinetics was found to fit experimental conversion-time data accurately. The activation energies of the main reaction and catalyst deactivation were obtained as 121 and −85 kJ/mol, respectively. The apparent negative value of the latter was attributed to a predominant deactivation mechanism via exothermic reversible partial hydration of active γ-alumina phase to the less active boehmite. At higher reaction temperatures, however, this mechanism is ceased at the expense of irreversible sintering of alumina crystallites.

Kinetic insights on wet peroxide oxidation of caffeine using EDTA-functionalized low-cost catalysts prepared from compost generated in municipal solid waste treatment facilities

Nowadays, sorted organic fraction of municipal solid waste is typically treated by anaerobic digestion processes, resulting therein a solid stream, further processed to obtain compost, whose production is higher than the existing demand as fertilizer. The current work proposes an alternative strategy for the recovering of compost through the production of low-cost catalysts by calcination (1073 K) and sulfuric acid treatments, followed by sequential functionalization with tetraethyl orthosilicate (TEOS) and ethylenediamine tetraacetic acid (EDTA). Activity and stability of the catalysts are assessed in the wet peroxide oxidation of synthetic wastewater effluents contaminated with caffeine, a model micro-pollutant, achieving its complete removal after 6 h at 353–383 K and catalyst loads of 0.5–2.5 g L−1. The increase of the catalytic activity of the materials upon functionalization with TEOS and EDTA is demonstrated and a kinetic modeling of caffeine degradation and hydrogen peroxide consumption with the best catalyst is assessed by pseudo-first power-law rate equations.

Anomalous Nonlinear Dynamics Behavior of Fractional Viscoelastic Beams.

Fractional models and their parameters are sensitive to intrinsic microstructural changes in anomalous materials. We investigate how such physics-informed models propagate the evolving anomalous rheology to the nonlinear dynamics of mechanical systems. In particular, we study the vibration of a fractional, geometrically nonlinear viscoelastic cantilever beam, under base excitation and free vibration, where the viscoelasticity is described by a distributed-order fractional model. We employ Hamilton's principle to obtain the equation of motion with the choice of specific material distribution functions that recover a fractional Kelvin-Voigt viscoelastic model of order α. Through spectral decomposition in space, the resulting time-fractional partial differential equation reduces to a nonlinear time-fractional ordinary differential equation, where the linear counterpart is numerically integrated through a direct L1-difference scheme. We further develop a semi-analytical scheme to solve the nonlinear system through a method of multiple scales, yielding a cubic algebraic equation in terms of the frequency. Our numerical results suggest a set of α-dependent anomalous dynamic qualities, such as far-from-equilibrium power-law decay rates, amplitude super-sensitivity at free vibration, and bifurcation in steady-state amplitude at primary resonance.

A mechanistic evolutionary model explains the time-dependent pattern of substitution rates in viruses

SummaryEstimating viral timescales is fundamental in understanding the evolutionary biology of viruses. Molecular clocks are widely used to reveal the recent evolutionary histories of viruses but may severely underestimate their longer-term origins because of the inverse correlation between inferred rates of evolution and the timescale of their measurement. Here, we provide a predictive mechanistic model that readily explains the rate decay phenomenon over a wide range of timescales and recapitulates the ubiquitous power-law rate decay with a slope of −0.65. We show that standard substitution models fail to correctly estimate divergence times once the most rapidly evolving sites saturate, typically after hundreds of years in RNA viruses and thousands of years in DNA viruses. Our model successfully recreates the observed pattern of decay and explains the evolutionary processes behind the time-dependent rate phenomenon. We then apply our model to re-estimate the date of diversification of genotypes of hepatitis C virus to 423,000 (95% highest posterior density [HPD]: 394,000–454,000) years before present, a time preceding the dispersal of modern humans out of Africa, and show that the most recent common ancestor of sarbecoviruses dates back to 21,000 (95% HPD: 19,000–22,000) years ago, nearly thirty times older than previous estimates. This creates a new perspective for our understanding of the origins of these viruses and also suggests that a substantial revision of evolutionary timescales of other viruses can be similarly achieved.

Spectropolarimetry and photometry of the early afterglow of the gamma-ray burst GRB 191221B

ABSTRACT We report on results of spectropolarimetry of the afterglow of the long gamma-ray burst GRB 191221B, obtained with SALT/RSS and VLT/FORS2, as well as photometry from two telescopes in the MASTER Global Robotic Network, at the MASTER-SAAO (South Africa) and MASTER-OAFA (Argentina) stations. Prompt optical emission was detected by MASTER-SAAO 38 s after the alert, which dimmed from a magnitude (white-light) of ∼10–16.2 mag over a period of ∼10 ks, followed by a plateau phase lasting ∼10 ks and then a decline to ∼18 mag after 80 ks. The light curve shows complex structure, with four or five distinct breaks in the power-law decline rate. SALT/RSS linear spectropolarimetry of the afterglow began ∼2.9 h after the burst, during the early part of the plateau phase of the light curve. Absorption lines seen at ∼6010 and 5490 Å are identified with the Mg ii 2799 Å line from the host galaxy at z = 1.15 and an intervening system located at z = 0.96. The mean linear polarization measured over 3400–8000 Å was ∼1.5 per cent and the mean equatorial position angle (θ) was ∼65°. VLT/FORS2 spectropolarimetry was obtained ∼10 h post-burst, during a period of slow decline (α = −0.44), and the polarization was measured to be p = 1.2 per cent and θ = 60°. Two observations with the MeerKAT radio telescope, taken 30 and 444 d after the GRB trigger, detected radio emission from the host galaxy only. We interpret the light curve and polarization of this long GRB in terms of a slow-cooling forward shock.

Normalization of the power-law rate equation for determining the kinetic parameters of the char-CO2 reaction

The power-law (or nth-order) rate equation is the most commonly used model to describe the kinetics of the char−CO2 reaction, which consists of three major kinetic parameters: the activation energy (E), the pre-exponential factor (A0), and the reaction order (n). So far, the temperature dependence of the reaction order and its influence on determining kinetic parameters have not attracted enough attention. This paper specially addresses the impact of the CO2 partial pressure on the reaction kinetics. CO2 gasification of the lignite char and coffee grounds char is conducted via thermogravimetry analysis (TGA) at 1098–1173 K and 0.025–0.05 MPa CO2 partial pressure. An exploration is made to normalize the power-law rate equation for kinetic computation, and pressure-independent E and A0 are obtained. Findings and insights obtained in this study can offer reliable evaluation of kinetic parameters and comparability of different kinetic studies.

Modelling of Power-Law Fluid Flow Inside a Piezoelectric Inkjet Printhead.

Piezoelectric three-dimensional inkjet printing has been used to manufacture heterogeneous objects due to its high level of flexibility. The materials used are non-Newtonian inks with complex rheological properties, and their behavior in the context of inkjet printing has not been fully understood: for example, the fact that the shear-thinning viscosity affects the droplet generation. Therefore, a control strategy coping with shear-thinning behaviors is needed to ensure printing consistency. In this paper, a novel model-based approach is presented to describe the shear-thinning ink dynamics inside the piezoelectric inkjet printhead, which provides the basis to design the excitation parameters in a systematic way. The dynamic equation is simplified into a quasi-one-dimensional equation through the combination of the boundary layer theory and the constitutive equation of the power-law fluid, of which the viscosity is shear-thinning. Based on this, a nonlinear time-varying equivalent circuit model is presented to simulate the power-law fluid flow rate inside the tube. The feasibility and effectiveness of this model can be evaluated by comparing the results of computational fluid dynamics and the experimental results.

A Mechanistic Evolutionary Model Explains the Time-Dependent Pattern of Substitution Rates in Viruses

Estimating viral timescales is fundamental to understanding the evolutionary biology of viruses. Molecular clocks are widely used for revealing their recent evolutionary histories but may severely underestimate their longer-term origins. This is because inferred rates of evolution are inversely correlated to the timescale of their measurement1,2. This rate decay is well described by a power law with a slope of -0.65 and can be observed across all known viral genome types3, implying a common process. We provide the first predictive mechanistic model that can readily explain the pattern of rate decay over a wide range of timescales and recapitulates the ubiquitous power-law rate decay with a slope of -0.65 (95% HPD: -0.72, -0.52). We show that standard substitution models fail to correctly estimate divergence times once the most rapidly evolving sites saturate, typically after hundreds of years in RNA viruses and thousands of years in DNA viruses, whereas we can successfully re-create the observed pattern of decay and explain the evolutionary processes involved in creating the time-dependent rate phenomenon. We apply our model to re-date the diversification of genotypes of hepatitis C virus (HCV) to 396,000 (95% HPD: 326,000 - 425,000) years before present, a time preceding the dispersal of modern humans out of Africa, and also showed that the most recent common ancestor of sarbecoviruses dates back to 23,500 (95% HPD: 21,100 - 25,300) years ago, nearly thirty times older than previous estimates4. This not only creates a radical new perspective for our understanding the origins of these viruses, but also suggests a substantial revision of evolutionary timescales of other viruses can be similarly achieved.